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ON TWO-PHASE STEFAN PROBLEM ARISING FROM A MICROWAVE HEATING PROCESS

R.V. Manoranjan and Hong-Ming Yin Department of Mathematics Washington State University Pullman, WA 99164, USA

R. Showalter Department of Mathematics Oregon State University Corvallis, OR 97331, USA. USA (Communicated by Alain Miranville)

Abstract. In this paper we study a free boundary problem modeling a phasechange process by using microwave heating. The mathematical model consists of Maxwell’s equations coupled with nonlinear heat conduction with a phasechange. The enthalpy form is used to characterize the phase-change process in the model. It is shown that the problem has a global solution. 1. Introduction. Suppose that a material with solid and liquid phases occupies a bounded domain Ω ⊂ R3 with C 1 -boundary ∂Ω. If we supply heat to the material by using intense microwaves from the boundary ∂Ω, the solid phase of the material will begin to melt. To describe this physical process, we introduce the electric and magnetic fields E(x, t) and H(x, t), respectively, in Ω. Hereafter, a bold letter represents a vector or vector function in three-space dimensions. Then E(x, t) and H(x, t) satisfy the following well-known Maxwell equations (see [9]): εEt + σE = ∇ × H,

(x, t) ∈ QT ,

µHt + ∇ × E = 0,

(1.1)

(x, t) ∈ QT ,

(1.2)

where QT = Ω × (0, T ] and J(x, t) = σE is used by Ohm’s law, ε, µ and σ are the electric permittivity, magnetic permeability and the electric conductivity, respectively. Let u(x, t) be the temperature in QT . During the heating process, the local density of heat source generated by microwaves is equal to E · J = σ(x, u)|E|2 , where the electric conductivity σ = σ(x, u) typically depends on u such as (see [12, 13]) σ(x, u) =

a(x) , or σ(x, u) = a(x)e−b(x)u , b(x) + c(x)u

where a(x), b(x) and c(x) are positive functions. It also may have a jump discontinuity for the temperature changing from solid phase to liquid phase:  σl (x, u), if u(x, t) < m, σ(x, u) = σs (x, u), if u(x, t) > m, where the subscript l or s represents the function in liquid or solid phase and m is the melting temperature. By using the enthalpy method, we find that the temperature u(x, t) satisfies the following heat equation in the weak sense (see [4, 5] and also Remark 1.1 below ), A(u)t − ∇ · [k(x, u)∇u] = σ(x, u)|E|2 ,

(x, t) ∈ QT ,

2000 Mathematics Subject Classification. Primary: 35R35. Key words and phrases. Microwave heating, phase-change, global existence. .

1

(1.3)

2

V.S. MANORANJAN, R. SHOWALTER, HONG-MING YIN

where

 if u < m,  u − 1, A(u) = [m − 1, m], if u = m,  u, if u > m, and the coefficient of heat conduction k may be different in solid and liquid phases,  kl (x, u), if u(x, t) < m, k(x, u) = ks (x, u), if u(x, t) > m, Because of the heat source in Eq.(1.3), the interface set ΓT = {(x, t) ∈ QT : u(x, t) = m} may have positive area, i.e. a mushy region may exist. In this case one has to define the value of heat conductivity k(x, u) and σ(x, u) on ΓT and Eq.(1.2) is understood as an inclusion ([4]: A(u)t − ∇[k(x, u)∇u] − σ(x, u)|E|2 3 0,

(x, t) ∈ QT .

Define σ(x, m) is between the values σl (x, m+) and σs (x, m−) for any x ∈ ΓT , k(x, m) is between the values of kl (x, m+) and ks (x, m−) for any x ∈ ΓT , where σl (x, m+) = limu→m+ σl (x, u) and other quantities are defined similarly. If σ(x, u) and k(x, u) are independent of x. Then one can simply define σ(m) ∈ [σ0 , σ1 ], k(m) ∈ [k0 , k1 ], where constants k0 , k1 , σ0 and σ1 are defined as follows: k0

=

min{kl (m), ks (m)}, k1 = max{kl (m), ks (m)},

σ0

=

min{σl (m), σs (m)}, σ1 = max{σl (m), σs (m)}.

To complete the problem, we prescribe the following initial and boundary conditions: N × E(x, t) = N × G(x, t), un (x, t) = 0,

(x, t) ∈ ST

(1.4)

(x, t) ∈ ST ,

E(x, 0) = E0 (x), H(x, 0) = H0 (x), u(x, 0) = u0 (x),

(1.5) x ∈ Ω,

(1.6)

where G(x, t) is given external vector function on ST = ∂Ω × [0, T ], N is the outward normal on S = ∂Ω, un (x, t) := ∇u · N is the normal derivative on S, E0 (x), H0 (x) and u0 (x) are the prescribed initial electric, magnetic fields and initial temperature. The Stefan-type free boundary problems have been studied extensively by many researchers (see monographs [6, 11, 18] and many conference proceedings). The classical enthalpy method is widely used to describe a phase-change process (see [1, 3, 4, 5, 11, 15, 16] etc. for examples). For microwave heating problems without phase-change, some research has been carried out (see [7, 8, 12, 13, 19] etc. and also see recent lecture notes [22] Chapter 6 for the theory). When a phase-change takes place, Coleman [2] studied the microwave melting in one-space dimension and obtained some numerical solutions. In [17], Pangrie et al. used time-harmonic Maxwell’s equations and the enthalpy method to model the microwave melting process and obtained the numerical solution for a radially symmetric domain by using finite-difference method. In [20] the author studied a phase-change problem arising from microwave heating processes in one-space dimension, where a kinetic type condition is given on the interface due to the superheating phenomenon. Global existence and uniqueness are established in [20]. We would also like to mention a related work on a phase-change problem for the induction heating ([21]) in which displacement current is neglected and magnetic field is assumed to be time-harmonic. However, none of the previous works deal with the fully coupled system (1.1)-(1.3) with phase-change. One of the difficulties is that there is not much known about the regularity of solutions to Maxwell’s equations with variable coefficients. Another difficulty is the nonlinear term σ(x, u)|E|2 which only belongs to L1 (QT ). Moreover, the electric conductivity may have a jump discontinuity from solid to liquid phase. In this paper we study the phase-change problem (1.1)-(1.6). By using methods from [21] it is shown that under certain conditions on coefficients of the system (1.1)-(1.3) the problem (1.1)-(1.6) has a global weak solution. The global existence is also established for the case when the electric and magnetic fields are assumed to be time-harmonic. Moreover, for one-space dimension we prove the existence of a weak solution for σ(x, t, u) with linear growth in the u-variable. In this paper the uniqueness is left out as an open question even for space dimension one. This paper is organized as follows. In section 2, we prove that the problem (1.1)-(1.6) has a weak solution in QT for any T > 0. In section 3, we study the problem for time-harmonic electric and magnetic fields and obtain the global solution for the problem. In section 4, we study the

PHASE-CHANGE IN MICROWAVE HEATING

3

problem for one-space dimension and prove the existence of a weak solution for a more general function σ(x, u). 2. Global Existence of Weak Solutions. In this section we first define weak solution to the problem and then consider an approximate problem by the standard approximation for A(u) and σ(u). It is shown that the approximated problem has a unique solution. Moreover, some uniform estimates for the approximate solution are derived. Finally, we establish the global existence to the problem (1.1)-(1.5) by using a compactness argument. We list some basic assumptions for the coefficients and the known data. H(2.1): (a)Let ε(x) and µ(x) be in L∞ (Ω) with a positive lower bound 0 < r0 ≤ ε(x), µ(x) ≤ R0 ,

x ∈ Ω,

where r0 and R0 are positive constants. (b) σ(x, u) is non-negative and is bounded in Ω × [M, ∞) for some large M > 0 and σ0 : 0 ≤ σ(x, u) ≤ σ0 , uσ(x, u) ≤ σ0 ,

(x, u) ∈ Ω × [M, ∞).

C 1+α (Ω × R)

(c) The functions kl (x, u) and ks (x, u) are of class bound: 0 < r0 ≤ ks (x, u), kl (x, u) ≤ R0 ,

and bounded with a positive lower

(x, u) ∈ Ω × [0, ∞).

H(2.2): (a) Let u0 (x) be in ∈ L∞ (Ω) and E0 (x), H0 (x) ∈ L2 (Ω)3 . 1

(b) Let G(x, t) ∈ C([0, T ]; H 2 (S)). It is easy to see that the conditions on σ(x, u) are satisfied for σ(x, u) =

1 (1+u)p

with p ≥ 1 or

σ(x, u) = a(x)e−u . For the reader’s convenience, we recall some function spaces associated with Maxwell’s equations. Other Sobolev spaces are the same as in [10]. Let H(curl, Ω)

=

{V ∈ L2 (Ω)3 : ∇ × V ∈ L2 (Ω)3 };

H0 (curl, Ω)

=

{V ∈ H(curl, Ω) : N × V = 0 on ∂Ω}.

H(curl, Ω) is a Hilbert space equipped with inner product Z (V, K) = [V · K + (∇ × V) · (∇ × K)]dx. Ω

Definition 2.1: A triple of functions (E(x, t), H(x, t), u(x, t)) is said to be a weak solution to the problem (1.1)-(1.6), if E(x, t), H(x, t) ∈ C([0, T ]; L2 (Ω)), T and u(x, t) ∈ L2 (0, T ; H 1 (Ω)) C([0, T ]; L2 (Ω)), and they satisfy the following integral identities: Z TZ Z TZ Z [−εE · Ψt + σE · Ψ]dxdt = [H · (∇ × Ψ)]dxdt + εE0 · Ψ(x, 0)dx, 0

Z

Ω T

Z

T

Z

0

Ω

Ω

Z [−µH · Φt + E · (∇ × Φ)]dxdt =

0

Z

Ω

[µH0 (x) · Φ(x, 0)]dx, Ω T

Z

Z

σ(x, u)|E|2 ψdxdt +

[−A(u)ψt + k(x, u)∇u∇ψ]dxdt = 0

Ω

0

Ω

Z A(u0 )ψdx Ω

for any test vector functions Ψ, Φ ∈ L2 (0, T ; H0 (curl, Ω)) C([0, T ]; L2 (Ω)3 ) and any test function ψ ∈ H 1 (0, T ; H 1 (Ω)) with Ψ(x, T ) = Φ(x, T ) = 0 and ψ(x, T ) = 0 on Ω. Since the weak solution E(x, t) ∈ L2 (Ω), we have to specify the boundary condition (1.4) in the weak sense. Note that Z t 1 1 H(x, t) = H0 (x) − ∇ × E(x, τ )dτ = H0 (x) − ∇ × W(x, t), µ(x) 0 µ(x) T

where Z W(x, t) =

t

E(x, τ )dτ.

0 L2 (Ω)3

It follows that H ∈ L2 (Ω)3 implied ∇ × W ∈ for each a.e. fixed t ∈ [0, T ]. Consequently, the trace N × W(x, t) is well-defined on ∂Ω. We define N × (E(x, t) − G(x, t)) = 0,

(x, t) ∈ ST

4

V.S. MANORANJAN, R. SHOWALTER, HONG-MING YIN

if and only if t

Z N × [W(x, t) −

G(x, τ )dτ ] = 0,

(x, t) ∈ ST .

0

Introduce a new function, Z

u

k(x, s)ds,

U (x, t) := K(x, u) =

(x, t) ∈ QT .

m

Then the assumption H(2.1)(b) implies that the inverse function u(x, t) = K −1 (x, U ) exists. Consequently, Eq. (1.3) can be written in the weak sense as follows: A∗ (x, U )t − ∆U = σ ∗ (x, U )|E|2 , where

(x, t) ∈ QT ,

 −1  Ks (x, U ) − 1, A (x, U ) = [−1, 0],  Kl−1 (x, U ), ∗

if U < 0, if U = 0, if U > 0,

and Ks−1 (x, U ), Kl−1 (x, U ) are the inverse functions of Ks (x, u), Kl (x, u), respectively. Moreover,  −1  σ(x, Ks (x, U )), if U < 0, ∗ σ (x, U ) = [σ0 , σ1 ], if U = 0,  σ(x, Kl−1 (x, U )), if U > 0. ¿From now on, instead of using U (x, t), A∗ (x, U ) and σ ∗ (x, U ), we will continue to use notation u(x, t), A(x, u) and σ(x, u) for simplicity. By assumption H(2.2), there exists an extension for G(x, t) such that G(x, t) ∈ C([0, T ]; H 1 (Ω)3 ). Moreover, from the assumption H(2.1)(c) there exists a constant a0 > 0 such that A0 (x, u) := Au (x, u) ≥ a0 for all (x, u) ∈ Ω × R whenever u 6= m. Let An (x, u) and σn (x, u) be smooth approximations of A(x, u) and σ(x, u), respectively. Moreover, we require that 1 An (x, u) = A(x, u), σn (x, u) = σ(x, u), if |u − m| ≥ n , r 0 0 , An (x, u) → A(x, u), σn (x, u) → σ(x, u) An (x, u) ≥ 2

strongly in L2 (Ω × [−M, M ]) for some large M > 0 as n → ∞ . We also make a smooth approximation of u0 (x), denoted by u0n (x), such that ∇u0n (x) = 0 on S and u0n (x) → u0 (x) strongly in L2 (QT ). Consider the following approximate system: ε(x)Et + σn (x, u)E = ∇ × H,

(x, t) ∈ QT ,

(2.1)

µ(x)Ht + ∇ × H = 0,

(x, t) ∈ QT ,

(2.2)

An (x, u)t − ∆u = σn (x, u)|E|2 ,

(x, t) ∈ QT ,

N × E(x, t) = N × G(x, t),

(x, t) ∈ ST ,

(2.3) (2.4)

(x, t) ∈ ST ,

(2.5)

E(x, 0) = E0 (x), H(x, 0) = H0 (x), u(x, 0) = u0n (x), x ∈ Ω.

(2.6)

un (x, t) = 0,

¿From Theorem 2.1 ([19]), the problem (2.1)-(2.6) has a unique weak solution (En (x, t), Hn (x, t)) ∈ C([0, T ]; H0 (curl, Ω)) × C([0, T ]; H(curl, Ω)) and un (x, t) ∈ C([0, T ]; L2 (Ω))

\

L2 (0, T ; H 1 (Ω)).

Moreover, En (x, t) − G(x, t) ∈ L∞ (0, T ; H0 (curl, Ω)). Furthermore, since u0 (x) ∈ L∞ (Ω) and σn (x, u)|E|2 ≥ 0, it follows from the maximum principle that there exists a constant M0 > 0 independent of n such that un (x, t) ≥ −M0 on QT . Now we derive some uniform estimates. Lemma 2.1: There exists constant C1 such that Z sup [|En |2 + |Hn |2 ]dx ≤ C1 , 0≤t≤T

Ω

PHASE-CHANGE IN MICROWAVE HEATING

5

where C1 depends only on the known data. Proof: For simplicity, we shall drop the subscript n for the solution (En , Hn , un ) whenever without causing confusion. To derive the estimate, we take the inner product by E(x, t) − G to Eq. (2.1) and by H(x, t) to Eq. (2.2), respectively, to obtain: Z Z d 1 [ε|E|2 ]dx + σ(x, u)|E|2 dx dt 2 Ω Ω Z Z Z = εE · Gdx + [σE · G]dx + [∇ × H · (E − G)]dx, Ω Ω Ω Z Z d 1 [µ|H|2 ]dx + [∇ × E · H]dx = 0. dt 2 Ω Ω We add up the above equations and use the fact, Z Z ∇ × H · (E − G)dx = H · [∇ × (E − G)]dx, Ω

Ω

to obtain Z [ε|E|2 + µ|H|2 ]dx + σ(x, u)|E|2 dx ZΩ ZΩ ≤C [|E|2 + |H|2 ]dx + C [|G|2 + |∇ × G|2 ]dx, d 1 dt 2

Z

Ω

Ω

where C depends only on L∞ -bounds of ε(x), µ(x) and σ(x, u), but independent of n. Gronwall’s inequality yields the desired estimate. Q.E.D. Lemma 2.2: There exists a constant C2 such that Z Z Z sup |un |2 dx + 0≤t≤T

Ω

|∇un |2 dxdt ≤ C2 ,

QT

where C2 depends only on known data. Proof: Since A0n (x, u) ≥ r20 , the inverse of the function for v(x, t) := An (x, u) exists, denoted by u = Bn (x, v). Then Z tZ Z t Z Z v d Bn (x, s)dsdx]dt An (u)t udxdt = [ 0 Ω 0 dt Ω 0 Z Z An (u0 ) Z Z v = Bn (x, s)dsdx − Bn (x, s)dsdx Ω 0 Ω 0 Z ≥ b0 u2 dx − C, Ω

for some b0 > 0 since k0 s ≤ Bn (x, s) ≤ k1 (s + 1) from the assumption H(2.1)(c). On the other hand, it is clear that Z tZ Z tZ |∇u|2 dxdt. − (∆u)udxdt = 0

0

Ω

Ω

Moreover, by Lemma 2.1 and the assumption H(2.1) we have Z σ(x, u)u|E|2 dx ≤ C, Ω

where C depends only on known data. We sum up the above estimates to obtain Z Z u2 dx + Ω

t 0

Z

|∇u|2 dxdt ≤ C2 , Ω

where C2 depends only on known data. Q. E. D. To prove the existence of a weak solution for the problem (1.1)-(1.6), we need the following lemma from [19].

6

V.S. MANORANJAN, R. SHOWALTER, HONG-MING YIN

Lemma 2.3: Suppose σn (x, t) → σ(x, t) strongly in L2 (QT ). Let (En (x, t), Hn (x, t)) be the solution of the Maxwell equations: εEt + σn (x, t)E = ∇ × H,

(x, t) ∈ QT ,

µHt + ∇ × E = 0,

(x, t) ∈ QT ,

N × E = N × G,

(x, t) ∈ ∂Ω × (0, T ], x ∈ Ω.

E(x, 0) = E0 (x), H(x, 0) = H0 (x),

Let (E(x, t), H(x, t)) be the solution of the above Maxwell equations where σn (x, t) is replaced by σ(x, t). Then (En (x, t), Hn (x, t)) converges to (E(x, t), H(x, t)) strongly in L2 (QT ). Q.E.D. Theorem 2.4: The problem (1.1)-(1.6) possesses at least one weak solution in QT for any T > 0. Proof: From Lemma 2.1-2.2 and the weak compactness, we know, after extracting a subsequence if necessary, that En (x, t) → E(x, t), Hn (x, t) → H(x, t) in weak-* L∞ (0, T ; L2 (Ω)3 ), un (x, t) → u(x, t) weakly in L2 (0, T ; H 1 (Ω)). Moreover, by applying the result of Lemma 5.1 from [15] we see that un (x, t) converges to u(x, t) strongly in L2 (QT ) and almost everywhere in QT . We multiply Eq.(2.1) and Eq.(2.2) by test functions Ψ(x, t) and Φ(x, t), respectively, in H 1 (0, T ; H0 (curl, Ω)) to obtain Z TZ [−εEn · Ψt + σn (x, un )En · Ψ]dxdt 0

Ω T Z

Z = Z

Z

εE0 (x) · Ψ(x, 0)dx, Z [−µHn · Φt + En · (∇ × Φ)]dxdt = [µH0 (x) · Φ(x, 0)]dx. [Hn · (∇ × Ψ)dxdt +

0

Ω

Ω

Ω

Ω

After taking the limit as n → ∞, we see that (E, H) satisfies the integral identities in Definition 2.1 if we can prove Jn (x, t) := σn (x, un )En converges to J(x, t) = σ(x, u)E(x, t) weakly in L2 (QT ). We omit the proof here since it can be done by using the same technique as for a more complicated term σn (x, un )|En |2 below (see detailed proof below). Moreover, since En (x, t) − G(x, t) ∈ L∞ (0, T ; H0 (curl, Ω)) and Hn (x, t) ∈ L∞ ([0, T ]; L2 (Ω)), it follows that ∇ × Wn (x, t) ∈ L∞ ([0, T ], L2 (Ω)), where Z t Wn (x, t) = En (x, τ )dτ. 0

R Thus, the trace N × Wn is well defined on ∂Ω. Since N × Wn = N × 0t G(x, τ )dτ and N × [En − Rt G] = 0 is equivalent to N × [Wn − 0 G(x, τ )dτ ] = 0. It follows that the boundary condition (1.4) holds. For any small γ > 0 by Egorof’s theorem there exists a subset Q ⊂ QT with |QT \Q| < γ such that un (x, t) converges to u(x, t) uniformly on Q. Set Qγ = {(x, t) ∈ Q : |u(x, t) − m| > γ}. Then, for (x, t) ∈ Qγ , if n is sufficiently large, |un (x, t) − m| ≥

γ 1 ≥ . 2 n

On the other hand, for any (x, t) ∈ Q\Qγ |un − u| ≤ |un − m| + |u − m| ≤ 2γ, provided that n is large enough. Let φ be a smooth test vector function. Z Z [σn (x, un )|En |2 − σ(x, u)|E|2 ]φdxdt Q

Z ZT =

[σn (x, un )|En |2 − σ(x, u)|E|2 ]φdxdt +

Q

Z Z

[σn (x, un )|En |2 − σ(x, u)|E|2 ]φdxdt

QT \Q

:= I1 + I2 .

PHASE-CHANGE IN MICROWAVE HEATING

7

It is clear that I2 → 0 as γ → 0 since |QT \Q| < γ. Z Z I1 := [σ(x, un )|En |2 − σ(x, u)|E|2 ]φdxdt Qγ

Z Z

[σn (x, un )|En |2 − σ(x, u)|E|2 ]φdxdt := J1 + J2 .

+ Q\Qγ

Z Z |J1 |

≤

σ(x, un )[|En |2 − |E|2 ]|φ|dxdt| +

|

Z Z

Qγ

:=

|σ(x, un ) − σ(x, u)|E|2 ]|φ|dxdt

Qγ

J11 + J12 .

It is clear that J11 → 0 since En converges to E strongly in L2 (QT ) by Lemma 2.3 and σ(x, un ) is bounded. On the other hand, \ [ \ Qγ = [Qγ {(x, t) ∈ Q : u(x, t) ≥ m + γ}] [Qγ {(x, t) ∈ Q : u(x, t) ≤ m − γ}]. T Since un → u(x, t) a.e. on QT and uniformly in Q, it follows that on Qγ {(x, t) : u(x, t) ≥ m+γ}, T σ(x, un ) = σl (x, un ) → σl (x, u) a.e. and on Qγ {(x, t) : u(x, t) ≤ m−γ}, σ(x, un ) = σs (x, un ) → σs (x, u) a.e. as n → ∞, by dominated convergence theorem we see J12 → 0 as n → ∞. Next we prove J2 → 0 as n → ∞. Without loss of generality, we assume σs (x, m) ≤ σl (x, m),

x ∈ Ω.

Then the approximation sequence σn (x, u) can be assumed to be increasing in u-vaiable in a neighborhood of u = m. On Q\Qγ , m − γ ≤ un (x, t) ≤ m + γ, As σn (x, u) is increasing in a neighborhood of u = m, we see σn (x, m − γ) ≤ σn (x, un ) ≤ σn (x, m + γ). If n is sufficiently large, σn (x, m − γ) = σs (x, m − γ), σn (x, m + γ) = σl (m + γ). By the weak convergence and the definition of σ(x, m), when γ → 0, n → ∞ we have σs (x, m−) ≤ σ(x, m) ≤ σ(x, m+),

a.e.x ∈ Ω.

It follows that Z Z |J2 |

≤

[σn (x, un )(|En |2 − |E|2 )|φ|dxdt|

| Q\Qγ

Z Z

(σn (x, un ) − σ(x, u)|E|2 ]φ)dxdt| → 0

+| Q\Qγ

as n → ∞, γ → 0. Next we consider the convergence for An (x, un ). The weak compactness implies that there exists a function β(x, t) ∈ L2 (QT ) such that An (un ) → β(x, t) weakly in L2 (QT ). We need to prove that the graph of β(x, t) ∈ A(u) a.e. on QT . From the construction of An and convergence of un we see β(x, t) = A(u) a.e. on Qγ . On QT \Qγ , m−

2 2 ≤ un ≤ m + . n n

The monotonicity of A(u) implies An (m −

2 2 ) ≤ An (un ) ≤ An (m + ), n n

which is the same as 2 2 ) ≤ An (un ) ≤ A(m + ). n n The weak convergence yields that for a.e. (x, t) ∈ QT \Qγ A(m −

m − 1 ≤ β(x, t) ≤ m.

8

V.S. MANORANJAN, R. SHOWALTER, HONG-MING YIN

Now we multiply Eq. (2.3) by any test function ψ ∈ H 1 (0, T ; H 1 (Ω)) with ψ(x, T ) = 0 and then integrate over QT to obtain Z TZ [−An (x, un )ψt + ∇un ∇ψ]dx 0

Ω T Z

Z

σ(x, un )|En |2 ψdxdt +

= 0

Z A(x, u0n (x))ψ(x, 0)dx.

Ω

Ω

Finally, we take limit as n → ∞ to see that (E(x, t), H(x, t), u(x, t)) is indeed a weak solution of the problem (1.1)-(1.6). Q.E.D. 3. Global Existence in Time-Harmonic Fields. For some industrial applications (see [12, 13]), the time scale for electromagnetic field and the heat conduction is quite different. It is often to assume that the electric and magnetic fields are time-harmonic. This leads to the following problem: iµωH + ∇ × E = 0,

x ∈ Ω,

(iεω + σ)E = ∇ × H,

(3.1)

x ∈ Ω,

(3.2)

where i represents the complex unit and ω is the frequency. For many applied problems, it is often convenient to use a unified approach by assuming that (see [12]): ε(x) = ε1 (x) + iε2 (x), µ(x) = µ1 (x) − iµ2 (x), where ε1 (x), ε2 (x), µ1 (x) and µ2 (x) are positive functions. It is clear that the system (3.1)-(3.2) is equivalent to the following one: ∇ × [γ(x)∇ × E] + r(x, u)E = 0,

x ∈ Ω,

(3.3)

where γ(x)

:=

r(x, u)

:=

1 µ2 µ1 = p + ip , µ(x) |µ1 |2 + |µ2 |2 |µ1 |2 + |µ2 |2 iω(iε(x)ω + σ(x, u)).

Consider the phase-change problem: ∇ × [γ(x)∇ × E] + r(x, u)E = 0, A(x, u)t − ∆u = σ(x, u)|E|2 , N × E(x) = N × G(x),

(3.4)

x ∈ ∂Ω,

(3.5) (3.6)

(x, t) ∈ ∂Ω × (0, T ],

un (x, t) = 0, u(x, 0) = u0 (x),

x ∈ Ω, (x, t) ∈ QT ,

x ∈ Ω.

(3.7) (3.8)

Note that the coefficient σ(x, u) depends on t since u(x, t) is a function of t. The solution E is also a function of t. However, this time variable for the heat conduction is different from the time-variable in electromagnetic field. A weak solution to (3.4)-(3.8) can be defined as follows. Definition 3.1: A pair functions (E(x, t), u(x, t)) is called a weak solution of (3.4)-(3.8) if E(x, t) ∈ H(curl, Ω), u(x, t) ∈ L2 (0, T ; H 1 (Ω)) with N × (E − G) ∈ H0 (curl, Ω) and the following integral identities hold: Z [γ(∇ × E) · (∇ × Ψ) + r(x, u)E · Ψ]dx = 0, ZΩZ Z Z Z [−A(x, u)ψt + ∇u∇ψ]dxdt = σ(x, u)|E|2 ψdxdt + A(x, u0 )ψ(x, 0)dx QT

ΩT

Ω

for any test functions Ψ, ∈ H0 (curl, Ω) and ψ ∈ H 1 (0, T ; H 1 (Ω)) with ψ(x, T ) = 0 on Ω. H(3.1): (a) Let ε1 (x), ε2 (x), µ1 (x) and µ2 (x) be nonnegative and of class L∞ (Ω) with ε1 ≥ r0 , ε2 ≥ 0. Let 0 ≤ σ(x, u) ≤ σ0 , uσ(x, u) ≤ σ1 , u ∈ [M, ∞). Moreover, there exists a constant σ1 such that σ(x, u) − |ε2 |L∞ (Ω) ≥ σ1 > 0.

PHASE-CHANGE IN MICROWAVE HEATING

9

(b) Let G(x) ∈ H(curl, Ω). H(3.2): Let A(x, u) be defined as in section 2 and satisfy the same condition as in H(2.1)(c). Moreover, u0 (x) ∈ L2 (Ω) and nonnegative. Theorem 3.1 Under the assumptions H(3.1)-(3.3), the problem (3.4)-(3.8) has a global weak solution. Proof: As the proof is quite similar to that of Theorem 2.4, we only give an outline. Step 1: Approximating the problem. By constructing smooth approximation for σ and A(x, u), we consider the approximation problem: ∇ × [γ(x)∇ × E] + rn (x, u)E = 0, An (x, u)t − ∆u = σn (x, u)|E|2 , N × E = N × G,

x ∈ Ω,

(3.9)

(x, t) ∈ QT ,

(3.10)

x ∈ ∂Ω,

(3.11)

(x, t) ∈ ∂Ω × (0, T ],

un (x, t) = 0, u(x, 0) = u0 (x),

x ∈ Ω.

(3.12) (3.13)

This problem has at least one weak solution (En (x, t), un (x, t)) ([21]). Step 2: Deriving uniform estimates. There exist constants C1 and C2 independent of n such that Z Z |∇ × En |2 dx + |En |2 dx ≤ C1 , Ω Ω Z Z Z sup u2n dx + |∇un |2 dxdt ≤ C2 . 0≤t≤T

Ω

QT

To prove the first estimate, we take the inner product by (E − G)∗ , the complex conjugate of E − G, to Eq.(3.9) to obtain Z γ(∇ × E) · [∇ × (E − G)∗ ] + rn (x, u)E · (E − G)∗ ]dx = 0. (3.14) Ω

We first take the imaginary part of the above equation to obtain Z Z µ2 q |∇ × E|2 dx + (σ − ε2 )|E|2 dx ≤ C, 2 2 Ω Ω µ 1 + µ2 where the constant C depends only on known data. By H(3.1), we obtain Z C |E|2 dx ≤ . σ1 Ω Now we take the real part of Eq. (3.15) and use the assumption H(3.1) again to obtain Z |∇ × E|2 dx ≤ C, Ω

where C depends only on known data. The second estimate is the same as Lemma 2.3. Step 3: Taking the limit. This step is almost identical to that of Theorem 2.4, we omit it here. Q.E.D. Remark 3.1: The assumption H(3.1) is only one type of sufficient condition to ensure the global existence of a unique weak solution to time-harmonic Maxwell’s equations. Other types of sufficient conditions can be found in [14, 22]. 4. One-dimensional Problem. In this section we study the problem (1.1)-(1.5) in one space dimension and prove that the weak solution exists globally for σ(x, t, u) with linear growth. Let QT = {(x, t) : 0 < x < 1, 0 < t < T }. For one-space dimension, we assume E(x, t) = {0, e(x, t), 0}, H(x, t) = {0, 0, h(x, t)}. Then the system (1.1)-(1.3) becomes the following form: ε(x)et + σ(x, t, u)e = −hx , µ(x)ht + ex = 0,

(x, t) ∈ QT , (x, t) ∈ QT , 2

A(x, u)t − uxx = σ(x, t, u)|e(x, t)| , where A(x, u) is the same as in Section 2.

(x, t) ∈ QT ,

(4.1) (4.2) (4.3)

10

V.S. MANORANJAN, R. SHOWALTER, HONG-MING YIN

By solving h(x, t) from Eq.(4.2), we see h(x, t) = h0 (x) −

1 wx (x, t), µ(x)

where

(x, t) ∈ QT ,

t

Z w(x, t) =

e(x, τ )dτ. 0

For simplicity, we assume h0 (x) = 0 on QT . It follows that Eq.(4.1)-(4.3) is equivalent to the following system: ε(x)wtt − (γ(x)wx )x + σ(x, t, u)wt = 0, A(x, u)t − uxx = σ(x, t, u)|wt (x, t)|2 ,

(4.4) (x, t) ∈ QT .

(4.5)

The initial and boundary conditions are as follows: w(0, t) = f1 (t), w(1, t) = f2 (t), ux (0, t) = ux (1, t) = 0, t ∈ [0, T ],

(4.6)

w(x, 0) = 0, wt (x, 0) = e0 (x), u(x, 0) = u0 (x),

(4.7)

0 < x < 1.

H(4.1): (a) Let ε(x), γ(x) satisfies H(2.1)(a). Let σ(x, t, u) satisfies 0 ≤ σ(x, t, u) ≤ σ0 (1 + u), (x, t, u) ∈ QT × [0, ∞). (b) Let A(x, u) be defined as in section 2 and satisfy the same condition as in H(2.1)(c). (c) Let f1 (t), f2 (t) ∈ H 1 (0, T ) with f1 (0) = f2 (0) = 0 and e0 (x), u0 (x) ∈ L2 (0, 1). Theorem 4.1: Under the assumption H(4.1), the problem (4.4)-(4.7) has a weak solution globally. Proof: As for n−dimensional case, we make a smooth approximation for σ(x, t, u) and A(x, u) and then consider the following approximate problem: ε(x)wtt − (γ(x)wx )x + σn (x, t, u)wt = 0, An (x, u)t − uxx = σn (x, t, u)|wt (x, t)|2 ,

(x, t) ∈ QT ,

(4.8)

(x, t) ∈ QT ,

(4.9)

w(0, t) = f1 (t), w(1, t) = f2 (t), ux (0, t) = ux (1, t) = 0, t ∈ [0, T ],

(4.10)

w(x, 0) = 0, wt (x, 0) = e0 (x), u(x, 0) = u0 (x),

(4.11)

0 < x < 1.

This problem has a unique weak solution ([19]) (wn (x, t), un (x, t)) ∈ H 1 (0, T ; H 1 (0, 1)) × L2 (0, T ; H 1 (0, 1)). Again we will omit the subscript n. Now we derive some uniform estimates. First of all, set g(x, t) = (1 − x)f1 (t) + xf2 (t), (x, t) ∈ QT . We multiply Eq.(4.8) by wt − gt (x, t) and then integrate over QT . Using the assumption H(4.1) and the growth condition for σ(x, t, u), we obtain, after some routine calculations, that Z 1 Z TZ 1 sup [wt2 + wx2 ]dx + σ(x, t, u)wt2 dxdt 0≤t≤T

0

0 T

Z

0

1

Z

≤ C 1 + C2

|u|dxdt, 0

0

where the constants C1 and C2 depend only on known data, but not on n. On the other hand, we use an estimate from the paper [3] to obtain Z 1 Z TZ 1 |u|dx ≤ C2 + C4 σ(x, t, u)|wt |2 dxdt, 0

0

0

where the constants C3 and C4 depend only on known data. It follows that Z 1 Z TZ 1 |u|dx ≤ C3 + C4 [C1 + C2 |u|dxdt]. 0

0

0

The above estimate holds if we replace T by any T 0 ∈ [0, T ], we can apply Gronwall’s inequality to obtain Z 1 |u|dx ≤ C5 , 0

where C5 depends only on known data.

PHASE-CHANGE IN MICROWAVE HEATING

11

Next, we multiply Eq. (4.9) by u and then integrate over QT 0 with T 0 ∈ (0, T ]. Using the same technique as in Section 2, we see Z Z Z An (x, u)t udxdt ≥ b0 u2 dx − C6 , QT 0

Ω

Z Z

Z Z uxx udxdt = −

QT 0

u2x dxdt

QT 0

where b0 > 0 and C6 depend only on known data. It follows that Z Z T0 Z 1 Z 1 u2x dxdt ≤ C + u2 (x, T 0 )dx + 0 T

Z ≤C+C 0

1

Z

|u|σ(x, t, u)wt2 dxdt

0

0

0

0

T

||u||2L∞ (0,1) dt.

Now, by Sobolev’s embedding ([10]), ||u||2L∞ (0,1)

≤ ≤

C||u||W 1,2 (0,1) ||u||L2 (0,1) Z 1 Z δ [u2 + u2x ]dx + C(δ) 0

1

u2 dx.

0

We combine the above estimates and choose δ sufficiently small to obtain Z 1 Z T0 Z 1 Z T0 Z 1 u2 dx + u2x dxdt ≤ C + C u2 dxdt. 0

0

0

0

0

Again Gronwall’s inequality yields 1

Z 0

u2 dx +

T0

Z 0

1

Z

u2x dxdt ≤ C8 ,

0

where the constant C8 depends only on known data. With those uniform estimates, as for Theorem 2.4 we can extract a subsequence from (wn (x, t), un (x, t)) and then take the limit to obtain a weak solution to the problem (4.4)-(4.7). We shall not repeat these steps here. Q.E.D. Remark 4.1: It would be interesting to show that the temperature is continuous over QT (see [1, 4]). Remark 4.2: The uniqueness is an open question, even for one-space dimension. REFERENCES [1] L. Caffarelli and L. C. Evans, Continuity of the temperature in the two-phase Stefan problem, in Free Boundary Problems: Theory and Applications, Res. Notes in Math., 78, Pitman, Boston, MA. 1983. [2] C.J. Coleman, The microwave heating of frozen substances, Appl. Mathe. Modelling, 14(1990), 439-440. [3] A. Damlamian, Some results on the multiphase Stefan problem, Comm. in P.D.Es, 2(1977), 1017-1044. [4] E. DiBenedetto and V. Vespri, Continuity for bounded solutions of multiphase Stefan problem, Atti Accad. Noz. Lincei CL. SCI. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 5(1994), 297-302. [5] A. Friedman, The Stefan problem in several space variables, Trans. AMS, 132(1968), 51-87. [6] A. Friedman, Variational Principles and Free Boundary Problems, John Wiley, New York, 1982. [7] R. Glassey and Hong-Ming Yin, On Maxwell’s equations with a temperature effect, II, Communications in Mathematical Physics, Vol. 194(1998), 343-358. [8] G.A. Kriegsmann, Microwave heating of dispersive media, SIAM J. App. Math., 53(1993), 655-669. [9] L. D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, New York, 1960. [10] O.A. La dyzenskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasi–linear Equations of Parabolic Type, AMS Trans. 23, Providence., R.I., 1968.

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V.S. MANORANJAN, R. SHOWALTER, HONG-MING YIN

[11] A.M. Meirmanov, The Stefan Problem, Walter de gruyter, Berlin, 1992. [12] A.C. Metaxas, Foundations of Electroheat, a unified approach, John Wiley & Sons, New York, 1996. [13] A.C. Metaxas and R.J. Meredith, Industrial Microwave Heating, I.E.E. Power Engineering Series,Vol. 4, Per Peregrimus Ltd., London, 1983. [14] C. M¨ uller, Foundations of the Mathematical Theory of Electromagnetic Waves, SpringerVerlag, New York, 1969. [15] M. Niezgodka and I. Pawlow, A generalized Stefan problem in several space variables, Applied Mathematics and Optimization, 9(1983), 193-224. [16] O.A. Oleinik, A method of solution of the general Stefan problem, Soviet Math. Dokl., 1(1960), 1350-1354. [17] B.J. Panrie, K.G. Ayappa, H.T. Davis, E.A. Davis and J. Gordon, Microwave thawing of cylinders, A.I.Ch.E. Journal, 31(1991), 1789-1800. [18] L.I. Rubinstein, The Stefan Problem, AMS translation, vol. 27, 1971. [19] H.M. Yin, On Maxwell’s equations in an electromagnetic field with the temperature effect, SIAM J. of Mathematical Analysis, 29(1998), 637-651. [20] H.M. Yin, On a free boundary problem with superheating arising in microwave heating processes, Advances in Math. Sci. Appl., 12(2002), 409-433. [21] H. M. Yin, Regularity of weak solution to Maxwell’s equations and applications to microwave heating, Journal of Differential Equations, 200(2004), 137-161. [22] H.M. Yin, Maxwell’s Equations in Electromagnetic Fields, Lecture Notes, Washington State University, Pullman, Washington, 2004. Received September 15, 2004; revised February 2005. E-mail address: [email protected]; [email protected]; [email protected]